Publications

Publications


  • Foreword: Special issue of JSC on the occasion of MEGA 2019 2022

    Bernardi, A.; D'Andrea, C.; Theobald, T., "Foreword: Special issue of JSC on the occasion of MEGA 2019" in JOURNAL OF SYMBOLIC COMPUTATION, v. 2022, n. 109 (2022), p. 199-201. - URL: https://www.sciencedirect.com/science/article/pii/S0747717120300560 . - DOI: 10.1016/j.jsc.2020.07.001

    2022 journal paper

  • Strict inclusions of high rank loci 2022

    Ballico, Edoardo; Bernardi, Alessandra; Ventura, Emanuele, "Strict inclusions of high rank loci" in JOURNAL OF SYMBOLIC COMPUTATION, v. 2022, n. 109 (2022), p. 238-249. - URL: https://www.sciencedirect.com/science/article/pii/S0747717120300596 . - DOI: 10.1016/j.jsc.2020.07.004

    For a given projective variety $X$, the high rank loci are the closures ofthe sets of points whose $X$-rank is higher than the generic one. We showexamples of strict inclusion between two consecutive high rank loci. Our firstexample is for the Veronese surface of plane quartics. Although Piene hadalready shown an example when $X$ is a curve, we construct infinitely manycurves in $mathbb P^4$ for which such strict inclusion appears. For spacecurves, we give two criteria to check whether the locus of points of maximalrank 3 is finite (possibly empty).

    2022 journal paper

  • SKEW-SYMMETRIC TENSOR DECOMPOSITION 2021

    Esteban Arrondo, Enrique; Bernardi, Alessandra; Macias Marques, Pedro; Mourrain, Bernardi, "SKEW-SYMMETRIC TENSOR DECOMPOSITION" in COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, v. 23, n. 2 (2021), p. 195006101-195006129. - URL: https://www.worldscientific.com/doi/10.1142/S0219199719500615 . - DOI: 10.1142/S0219199719500615

    2021 journal paper

  • Geometric conditions for strict submultiplicativity of rank and border rank 2021

    Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio; Oneto, Alessandro; Ventura, Emanuele, "Geometric conditions for strict submultiplicativity of rank and border rank" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 200, n. 1 (2021), p. 187-210. - URL: https://link.springer.com/article/10.1007/s10231-020-00991-6 . - DOI: 10.1007/s10231-020-00991-6

    2021 journal paper

  • High order singular value decomposition for plant diversity estimation 2021

    Bernardi, Alessandra; Iannacito, Martina; Rocchini, Duccio, "High order singular value decomposition for plant diversity estimation" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2021, n. 14(4) (2021), p. 557-591. - URL: https://doi.org/10.1007/s40574-021-00300-w . - DOI: 10.1007/s40574-021-00300-w

    2021 journal paper

  • rasterdiv—An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back 2021

    Rocchini, Duccio; Thouverai, Elisa; Marcantonio, Matteo; Iannacito, Martina; Da Re, Daniele; Torresani, Michele; Bacaro, Giovanni; Bazzichetto, Manuele; Bernardi, Alessandra; Foody, Giles M.; Furrer, Reinhard; Kleijn, David; Larsen, Stefano; Lenoir, Jonathan; Malavasi, Marco; Marchetto, Elisa; Messori, Filippo; Montaghi, Alessandro; Moudrý, Vítězslav; Naimi, Babak; Ricotta, Carlo; Rossini, Micol; Santi, Francesco; Santos, Maria J.; Schaepman, Michael E.; Schneider, Fabian D.; Schuh, Leila; Silvestri, Sonia; Ŝímová, Petra; Skidmore, Andrew K.; Tattoni, Clara; Tordoni, Enrico; Vicario, Saverio; Zannini, Piero; Wegmann, Martin, "rasterdiv—An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back" in METHODS IN ECOLOGY AND EVOLUTION, v. 2021, 12, n. 6 (2021), p. 1093-1102. - URL: https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041-210X.13583 . - DOI: 10.1111/2041-210X.13583

    2021 journal paper

  • Identifiability of Rank-3 Tensors 2021

    Ballico, Edoardo; Bernardi, Alessandra; Santarsiero, Pierpaola, "Identifiability of Rank-3 Tensors" in MEDITERRANEAN JOURNAL OF MATHEMATICS, v. 18, n. 4 (2021). - DOI: 10.1007/s00009-021-01788-4

    2021 journal paper

  • Waring, tangential and cactus decompositions 2020

    Bernardi, Alessandra; Taufer, Daniele, "Waring, tangential and cactus decompositions" in JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, v. 143, (2020), p. 1-30. - URL: https://www.sciencedirect.com/science/article/abs/pii/S0021782420301203 . - DOI: 10.1016/j.matpur.2020.07.003

    2020 journal paper

  • On the ranks of the third secant variety of Segre-Veronese embeddings 2019

    Ballico, Edoardo; Bernardi, Alessandra, "On the ranks of the third secant variety of Segre-Veronese embeddings" in LINEAR & MULTILINEAR ALGEBRA, v. 2019, 67, n. 3 (2019), p. 583-597. - URL: https://www.tandfonline.com/doi/abs/10.1080/03081087.2018.1430117 . - DOI: 10.1080/03081087.2018.1430117

    2019 journal paper

  • A note on the cactus rank for Segre-Veronese varieties 2019

    Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio, "A note on the cactus rank for Segre-Veronese varieties" in JOURNAL OF ALGEBRA, v. 526, (2019), p. 6-11. - URL: https://www.sciencedirect.com/science/article/pii/S002186931930078X . - DOI: 10.1016/j.jalgebra.2019.01.027

    We give an upper bound for the cactus rank of any multi-homogeneous polynomial.

    2019 journal paper

  • On the partially symmetric rank of tensor products of W-states and other symmetric tensors 2019

    Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias; Gesmundo, Fulvio, "On the partially symmetric rank of tensor products of W-states and other symmetric tensors" in ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI, v. 30, n. 1 (2019), p. 93-124. - URL: http://www.ems-ph.org/journals/journal.php?jrn=rlm . - DOI: 10.4171/RLM/837

    Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called emph{$W$-states}, namely monomials of the form $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 -1}y ootimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ cdots +d_k)$.

    2019 journal paper

  • Introduction 2019

    Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter, "Introduction" in "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, edited by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, M. Joseph Landsberg, Davide Pastorello, Bassano Vacchini, Frédéric Holweck, Luca Chiantini, F. M. Ciaglia, A. Ibort, G. Marmo, Switzerland: Springer, Cham, 2019, p. 4-9. - URL: https://link.springer.com/chapter/10.1007/978-3-030-06122-7_1 . - DOI: 10.1007/978-3-030-06122-7_1

    2019 part of book

  • Quantum Physics and Geometry 2019

    Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter (edited by), "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, Luca Chiantini, Frédéric Holweck, M. Joseph Landsberg, F.M. Ciaglia, Alberto Ibort, Giuseppe Marmo, Davide Pastorello, Bassano Vacchini, Switzerland: Springer, 2019, 172 p. - (LECTURE NOTES OF THE UNIONE MATEMATICA ITALIANA). - ISBN: 978-3-030-06121-0. - URL: https://www.springer.com/gp/book/9783030061210 . - DOI: 10.1007/978-3-030-06122-7

    2019 book

  • On polynomials with given Hilbert function and applications 2018

    Bernardi, Alessandra; Jelisiejew, Joachim; Macias Marques, Pedro; Ranestad, Kristian, "On polynomials with given Hilbert function and applications" in COLLECTANEA MATHEMATICA, v. 2018, n. Volume 69, Issue 1 (2018), p. 39-64. - URL: https://link.springer.com/article/10.1007/s13348-016-0190-2 . - DOI: 10.1007/s13348-016-0190-2

    Using Macaulay’s correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

    2018 journal paper

  • Typical and Admissible ranks over fields 2018

    Ballico, Edoardo; Bernardi, Alessandra, "Typical and Admissible ranks over fields" in RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, v. 67, n. 1 (2018), p. 115-128. - URL: https://link.springer.com/article/10.1007/s12215-017-0299-5?wt_mc=Internal.Event.1.SEM.ArticleAuthorAssignedToIssue . - DOI: 10.1007/s12215-017-0299-5

    Let $X(mathbb{R})$ be a geometrically connected variety defined over $mathbb{R}$ such that the set of all its complex points $X(mathbb{C})$ is non-degenerate. We introduce the notion of emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set $Ssubset X(mathbb{C})$ of points such that $S$ spans $P$ and $S$ is stable under conjugation. Any set evincing the admissible rank can be equipped with a emph{label} keeping track of the number of its complex and real points. We show that, in the case of generic identifiability, there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if $X$ is a rational normal curve then there always exists a label for the generic element. We present two examples in which either the label doesn't exist or the admissible rank is strictly bigger than the usual complex rank.

    2018 journal paper

  • Singularities of plane rational curves via projections 2018

    Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica, "Singularities of plane rational curves via projections" in JOURNAL OF SYMBOLIC COMPUTATION, v. 86, n. May–June (2018), p. 189-214. - URL: http://dx.doi.org/10.1016/j.jsc.2017.05.003 . - DOI: 10.1016/j.jsc.2017.05.003

    We consider the parameterization ${mayhbf{f}}=(f_0:,f_1:f_2)$ of a plane rational curve $C$ of degree $n$, and we study the singularities of $C$ via such parameterization. We use the projection from the rational normal curve $C_nsubsetmathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $mathbf{f}$ certain 0-dimensioal schemes $X_ksubset mathbb{P}^k$, $2leq k leq (n-1)$, which encode all information on the singularities of multiplicity $geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.

    2018 journal paper

  • On real typical ranks 2018

    Grigoriy, Blekherman; Ottaviani, Giorgio; Bernardi, Alessandra, "On real typical ranks" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2018, 11, n. 3 (2018), p. 293-307. - URL: https://link.springer.com/article/10.1007/s40574-017-0134-0 . - DOI: 10.1007/s40574-017-0134-0

    2018 journal paper

  • A new class of non-identifiable skew symmetric tensors 2018

    Bernardi, Alessandra; Vanzo, Davide, "A new class of non-identifiable skew symmetric tensors" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, n. Volume 197, Issue 5 (2018), p. 1499-1510. - URL: https://link.springer.com/article/10.1007/s10231-018-0734-z?wt_mc=alerts.TOCjournals&utm_source=toc&utm_medium=email&utm_campaign=toc_10231_197_5 . - DOI: 10.1007/s10231-018-0734-z

    We prove that the generic element of the fifth secant variety $sigma_5(Gr(mathbb{P}^2,mathbb{P}^9)) subset mathbb{P}(igwedge^3 mathbb{C}^{10})$ of the Grassmannian of planes of $mathbb{P}^9$ has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. {We show that this, {together with $Gr(mathbb{P}^3, mathbb{P}^8)$, is the only non-identifiable case} among the non-defective secant varieties $sigma_s(Gr(mathbb{P}^k, mathbb{P}^n))$ for any $n<14$. In the same range for $n$, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians.} We also show that the dual variety $(sigma_3(Gr(mathbb{P}^2,mathbb{P}^7)))^{ee}$ of the variety of 3-secant planes of the Grassmannian of $mathbb{P}^2subset mathbb{P}^7$ is $sigma_2(Gr(mathbb{P}^2,mathbb{P}^7))$ the variety of bi-secant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8-th dimensional ``Hilbert'' space.

    2018 journal paper

  • On the dimension of contact loci and the identifiability of tensors 2018

    Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca, "On the dimension of contact loci and the identifiability of tensors" in ARKIV FÖR MATEMATIK, v. 56, n. 2 (2018), p. 265-283. - URL: http://www.intlpress.com/site/pub/pages/journals/items/arkiv/content/vols/0056/0002/a004/index.html . - DOI: 10.4310/ARKIV.2018.v56.n2.a4

    Let X⊂ℙr be an integral and non-degenerate variety. Set n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the expected) dimension (k+n−1)(n+1)−1

    2018 journal paper

  • Bounds on the tensor rank 2018

    Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca; Guardo, Elena, "Bounds on the tensor rank" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, 197, n. 6 (2018), p. 1771-1785. - URL: http://link.springer.com/article/10.1007/s10231-018-0748-6 . - DOI: 10.1007/s10231-018-0748-6

    2018 journal paper

  • The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition 2018

    Bernardi, Alessandra; Carlini, Enrico; Virginia Catalisano, Maria; Gimigliano, Alessandro; Oneto, Alessandro, "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition" in MATHEMATICS, v. 6, n. 12 (2018), p. 31401-31486. - URL: https://www.mdpi.com/2227-7390/6/12/314/htm . - DOI: 10.3390/math6120314

    We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

    2018 journal paper

  • Curvilinear schemes and maximum rank of forms 2017

    Ballico, E; Bernardi, Alessandra, "Curvilinear schemes and maximum rank of forms" in LE MATEMATICHE, v. 72, n. 1 (2017), p. 137-144. - URL: https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/1360/1015 . - DOI: 10.4418/2017.72.1.10

    We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$-th Veronese embedding of $\mathbb{P}^n$, whose span contains the projective class of $P$. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.

    2017 journal paper

  • A uniqueness result on the decompositions of a bi-homogeneous polynomial 2017

    Ballico, Edoardo; Bernardi, Alessandra, "A uniqueness result on the decompositions of a bi-homogeneous polynomial" in LINEAR & MULTILINEAR ALGEBRA, v. 2017, 65, n. 4 (2017), p. 677-698. - URL: https://www.tandfonline.com/doi/full/10.1080/03081087.2016.1202182 . - DOI: 10.1080/03081087.2016.1202182

    In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial $p$ (i.e. a partially symmetric tensor of $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex, finite dimensional vector spaces) if its rank with respect to the Segre-Veronese variety $S_{d_1,d_2}(V_1,V_2)$ is at most $\min \{d_1,d_2\}$. Such a polynomial may not have a unique minimal decomposition as $p=\sum_{i=1}^r\lambda_i p_i$ with $p_i\in S_{d_1,d_2}(V_1,V_2)$ and $\lambda_i$ coefficients, but we can show that there exist unique $p_1, \ldots , p_{r'}$, $p_{1}', \ldots , p_{r''}'\in S_{d_1,d_2}(V_1,V_2) $, two unique linear forms $l\in V_1^*$, $l'\in V_2^*$, and two unique bivariate polynomials $q\in S^{d_2}V_2^*$ and $q'\in S^{d_1}V_1^*$ such that either $p=\sum_{i=1}^{r'} \lambda_i p_i+l^{d_1}q $ or $ p= \sum_{i=1}^{r''}\lambda'_i p_i'+l'^{d_2}q'$, ($\lambda_i, \lambda'_i$ being appropriate coefficients). In the second part of the paper we focus on the tangential variety of the Segre-Veronese varieties. We compute the rank of their tensors (that is valid also in the case of Segre-Veronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the two-factors Segre-Veronese varieties.

    2017 journal paper

  • Tensor decomposition and homotopy continuation 2017

    Bernardi, Alessandra; Noah, Daleo; Jonathan, Hauenstein; Bernard, Mourrain, "Tensor decomposition and homotopy continuation" in DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, v. 2017, n. 55 (2017), p. 78-105. - URL: https://www.sciencedirect.com/science/article/pii/S0926224517301055?via=ihub . - DOI: 10.1016/j.difgeo.2017.07.009

    2017 journal paper

  • On parameterizations of plane rational curves and their syzygies 2016

    Bernardi, Alessandra; Gimigliano, A.; Idà, M., "On parameterizations of plane rational curves and their syzygies" in MATHEMATISCHE NACHRICHTEN, v. 289, n. 5-6 (2016), p. 537-545. - URL: http://www3.interscience.wiley.com/journal/60500208/home . - DOI: 10.1002/mana.201500264

    Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ of $C$, where $mathcal{O}_{mathbb{P}^1}(-a-d)oplus mathcal{O}_{mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)subset K[s,t]$, and , $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2kleq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $PP^{k+1}$.

    2016 journal paper